-
Previous Article
Transversal intersections of invariant manifolds of NMS flows on $S^{3}$
- DCDS Home
- This Issue
-
Next Article
Lipschitz regularity of solution map of control systems with multiple state constraints
Uniqueness of equilibrium states for some partially hyperbolic horseshoes
1. | Instituto de Matemática - UFRJ, Av. Athos da Silveira Ramos 149, Cidade Universitária - Ilha do Fundão, P.O. Box 68530. Rio de Janeiro - RJ, Brazil, Brazil |
References:
[1] |
J. Alves and V. Araújo, Random perturbations of non-uniformly expanding maps, Astérisque, 286 (2003), 25-62. |
[2] |
J. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398.
doi: 10.1007/s002220000057. |
[3] |
A. Arbieto, C. Matheus and K. Oliveira, Equilibrium states for random non-uniformly expanding maps, Nonlinearity, 17 (2004), 581-593.
doi: 10.1088/0951-7715/17/2/013. |
[4] |
C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. of Math., 115 (2000), 157-193.
doi: 10.1007/BF02810585. |
[5] |
R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. |
[6] |
R. Bowen, Some systems with unique equilibrium states,, Math. Systems Theory, 8 (): 1974.
|
[7] |
R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.
doi: 10.1090/S0002-9947-1971-0274707-X. |
[8] |
R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphism," Springer Lecture Notes in Math., 470, 1975. |
[9] |
R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.
doi: 10.1007/BF01389848. |
[10] |
H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Comm. Math. Phys., 168 (1995), 571-580.
doi: 10.1007/BF02101844. |
[11] |
H. Bruin and G. Keller, Equilibrium states for S-unimodal maps, Ergodic Theory Dynam. Systems, 18 (1998), 765-789.
doi: 10.1017/S0143385798108337. |
[12] |
H. Bruin and M. Todd, Equilibrium states for interval maps: The potential $-t log\|Df\|$, Ann. Sci. École Norm. Sup., 42 (2009), 559-600. |
[13] |
J. Buzzi, Thermodynamical formalism for piecewise invertible maps: Absolutely continuous invariant measures as equilibrium states, Proc. Sympos. Pure Math., 69 (2001), 749-783. |
[14] |
J. Buzzi, T. Fisher, M. Sambarino and C. Vásquez, Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems,, Ergodic Theory and Dynamical Systems, ().
|
[15] |
J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps, Ergodic Theory Dynam. Systems, 23 (2003), 1383-1400.
doi: 10.1017/S0143385703000087. |
[16] |
L. J. Díaz and T. Fisher, Symbolic extensions for partially hyperbolic diffeomorphisms, Discrete and Continuous Dynamical Systems, 29 (2011), 1419-1441. |
[17] |
L. J. Díaz, V. Horita, M. Sambarino and I. Rios, Destroying horseshoes via heterodimensional cycles: Generating bifurcations inside homoclinic classes, Ergodic Theory and Dynamical Systems, 29 (2009), 433-474.
doi: 10.1017/S0143385708080346. |
[18] |
Haydn N.T.A. and D. Ruelle, Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification, Commun. Math. Phys., 148 (1992), 155-167.
doi: 10.1007/BF02102369. |
[19] |
F. Hofbauer, The topological entropy of a transformation $x\mapsto ax(1-x)$, Monatsh. Math., 90 (1980), 117-141.
doi: 10.1007/BF01303262. |
[20] |
G. Iommi and M. Todd, Natural equilibrium states for multimodal maps, Commun. Math. Phys., 300 (2010), 65-94.
doi: 10.1007/s00220-010-1112-x. |
[21] |
R. Israel, "Convexity in the Theory of Lattice Gases," Princeton University Press, 1979. |
[22] |
G. Keller, Lifting measures to Markov extensions, Monatsh. Math., 108 (1989), 183-200. |
[23] |
F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc., 16 (1977), 568-576.
doi: 10.1112/jlms/s2-16.3.568. |
[24] |
R. Leplaideur, K. Oliveira and I. Rios, Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes, Nonlinearity, 19 (2006), 2667-2694.
doi: 10.1088/0951-7715/19/11/009. |
[25] |
S. E. Newhouse, Continuity properties of entropy, Annals of Mathematics, 129 (1989), 215-235.
doi: 10.2307/1971492. |
[26] |
K. Oliveira, Equilibrium states for non-uniformly expanding maps, Ergodic Theory & Dynamical Systems, 23 (2003), 1891-1905.
doi: 10.1017/S0143385703000257. |
[27] |
Y. Pesin and S. Senti, Equilibrium measures for maps with inducing schemes, Journal of Modern Dynamics, 2 (2008), 397-430. |
[28] |
V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Russian Math. Surveys, 22 (1967), 3-56. |
[29] |
D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," Encyclopedia of Mathematics and its Applications, Addison-Wesley Publishing Co., Reading, Mass, 5, 1978. |
[30] |
P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps, Annales de l Institut Henri Poincaré. Analyse non Linéaire, 27 (2010), 555-593. |
[31] |
W. Cowieson and L. S. Young, SRB measures as zero-noise limits, Ergodic Theory Dynamic Systems, 25 (2005), 1115-1138.
doi: 10.1017/S0143385704000604. |
show all references
References:
[1] |
J. Alves and V. Araújo, Random perturbations of non-uniformly expanding maps, Astérisque, 286 (2003), 25-62. |
[2] |
J. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398.
doi: 10.1007/s002220000057. |
[3] |
A. Arbieto, C. Matheus and K. Oliveira, Equilibrium states for random non-uniformly expanding maps, Nonlinearity, 17 (2004), 581-593.
doi: 10.1088/0951-7715/17/2/013. |
[4] |
C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. of Math., 115 (2000), 157-193.
doi: 10.1007/BF02810585. |
[5] |
R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. |
[6] |
R. Bowen, Some systems with unique equilibrium states,, Math. Systems Theory, 8 (): 1974.
|
[7] |
R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.
doi: 10.1090/S0002-9947-1971-0274707-X. |
[8] |
R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphism," Springer Lecture Notes in Math., 470, 1975. |
[9] |
R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.
doi: 10.1007/BF01389848. |
[10] |
H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Comm. Math. Phys., 168 (1995), 571-580.
doi: 10.1007/BF02101844. |
[11] |
H. Bruin and G. Keller, Equilibrium states for S-unimodal maps, Ergodic Theory Dynam. Systems, 18 (1998), 765-789.
doi: 10.1017/S0143385798108337. |
[12] |
H. Bruin and M. Todd, Equilibrium states for interval maps: The potential $-t log\|Df\|$, Ann. Sci. École Norm. Sup., 42 (2009), 559-600. |
[13] |
J. Buzzi, Thermodynamical formalism for piecewise invertible maps: Absolutely continuous invariant measures as equilibrium states, Proc. Sympos. Pure Math., 69 (2001), 749-783. |
[14] |
J. Buzzi, T. Fisher, M. Sambarino and C. Vásquez, Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems,, Ergodic Theory and Dynamical Systems, ().
|
[15] |
J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps, Ergodic Theory Dynam. Systems, 23 (2003), 1383-1400.
doi: 10.1017/S0143385703000087. |
[16] |
L. J. Díaz and T. Fisher, Symbolic extensions for partially hyperbolic diffeomorphisms, Discrete and Continuous Dynamical Systems, 29 (2011), 1419-1441. |
[17] |
L. J. Díaz, V. Horita, M. Sambarino and I. Rios, Destroying horseshoes via heterodimensional cycles: Generating bifurcations inside homoclinic classes, Ergodic Theory and Dynamical Systems, 29 (2009), 433-474.
doi: 10.1017/S0143385708080346. |
[18] |
Haydn N.T.A. and D. Ruelle, Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification, Commun. Math. Phys., 148 (1992), 155-167.
doi: 10.1007/BF02102369. |
[19] |
F. Hofbauer, The topological entropy of a transformation $x\mapsto ax(1-x)$, Monatsh. Math., 90 (1980), 117-141.
doi: 10.1007/BF01303262. |
[20] |
G. Iommi and M. Todd, Natural equilibrium states for multimodal maps, Commun. Math. Phys., 300 (2010), 65-94.
doi: 10.1007/s00220-010-1112-x. |
[21] |
R. Israel, "Convexity in the Theory of Lattice Gases," Princeton University Press, 1979. |
[22] |
G. Keller, Lifting measures to Markov extensions, Monatsh. Math., 108 (1989), 183-200. |
[23] |
F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc., 16 (1977), 568-576.
doi: 10.1112/jlms/s2-16.3.568. |
[24] |
R. Leplaideur, K. Oliveira and I. Rios, Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes, Nonlinearity, 19 (2006), 2667-2694.
doi: 10.1088/0951-7715/19/11/009. |
[25] |
S. E. Newhouse, Continuity properties of entropy, Annals of Mathematics, 129 (1989), 215-235.
doi: 10.2307/1971492. |
[26] |
K. Oliveira, Equilibrium states for non-uniformly expanding maps, Ergodic Theory & Dynamical Systems, 23 (2003), 1891-1905.
doi: 10.1017/S0143385703000257. |
[27] |
Y. Pesin and S. Senti, Equilibrium measures for maps with inducing schemes, Journal of Modern Dynamics, 2 (2008), 397-430. |
[28] |
V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Russian Math. Surveys, 22 (1967), 3-56. |
[29] |
D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," Encyclopedia of Mathematics and its Applications, Addison-Wesley Publishing Co., Reading, Mass, 5, 1978. |
[30] |
P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps, Annales de l Institut Henri Poincaré. Analyse non Linéaire, 27 (2010), 555-593. |
[31] |
W. Cowieson and L. S. Young, SRB measures as zero-noise limits, Ergodic Theory Dynamic Systems, 25 (2005), 1115-1138.
doi: 10.1017/S0143385704000604. |
[1] |
Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 |
[2] |
Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192 |
[3] |
Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435 |
[4] |
Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673 |
[5] |
Vaughn Climenhaga, Yakov Pesin, Agnieszka Zelerowicz. Equilibrium measures for some partially hyperbolic systems. Journal of Modern Dynamics, 2020, 16: 155-205. doi: 10.3934/jmd.2020006 |
[6] |
Lorenzo J. Díaz, Todd Fisher, M. J. Pacifico, José L. Vieitez. Entropy-expansiveness for partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4195-4207. doi: 10.3934/dcds.2012.32.4195 |
[7] |
Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469 |
[8] |
Vítor Araújo. Semicontinuity of entropy, existence of equilibrium states and continuity of physical measures. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 371-386. doi: 10.3934/dcds.2007.17.371 |
[9] |
Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341 |
[10] |
Radu Saghin. Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3789-3801. doi: 10.3934/dcds.2014.34.3789 |
[11] |
Suzete Maria Afonso, Vanessa Ramos, Jaqueline Siqueira. Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4485-4513. doi: 10.3934/dcds.2021045 |
[12] |
Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215 |
[13] |
Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266 |
[14] |
Xinsheng Wang, Weisheng Wu, Yujun Zhu. Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 81-105. doi: 10.3934/dcds.2020004 |
[15] |
Yun Yang. Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5133-5152. doi: 10.3934/dcds.2015.35.5133 |
[16] |
Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020 |
[17] |
Omri M. Sarig. Bernoulli equilibrium states for surface diffeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 593-608. doi: 10.3934/jmd.2011.5.593 |
[18] |
Dominic Veconi. Equilibrium states of almost Anosov diffeomorphisms. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 767-780. doi: 10.3934/dcds.2020061 |
[19] |
Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037 |
[20] |
Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]